On the Brauer liftings for modular representations
For any finite group G, if \(\rho\) : \(G\to GL(n,K)\) is a representation of G over a field K of characteristic \(p>0\), it is known [\textit{J. A. Green}, Trans. Am. Math. Soc. 80, 402-447 (1955; Zbl 0068.25605)] that \(\rho\) is the reduction mod \({\mathfrak p}\) (\({\mathfrak p}\) an appropriate prime ideal) of a virtual representation of G over \({\mathbb{C}}\). The latter is called the ''Brauer lift'' Br(\(\rho)\) of \(\rho\). \textit{G. Lusztig} [The discrete series of \(GL_ n\) over a finite field (Ann. Math. Stud. 81, 1974; Zbl 0293.20038)] gave an explicit expression for Br(\(\rho)\), where \(\rho\) is the defining representation of \(GL(n,{\mathbb{F}}_ q)\) \(({\mathbb{F}}_ q\) being the field of \(q=p^ r\) elements). Lusztig produced explicit irreducible representations \(X_ 1,...,X_ n\) whose characters were known such that \(Br(\rho)=X_ 1-X_ 2+...+(-1)^{n-1}X_ n.\) In the work under review, the author gives the irreducible decomposition of Br(\(\rho)\), where \(\rho\) is the highest weight module of an \({\mathbb{F}}_ q\)-split Chevalley group \(G^ F\) which corresponds to a ''small'' weight \(\chi\). His method is derived from that of Lusztig: he gives a decomposition of the projective variety G/P(\(\chi)\) (where P(\(\chi)\) is the parabolic subgroup of the algebraic group G which corresponds to the stabilizer W(\(\chi)\) of \(\chi\) in the Weyl group W) into locally closed subvarieties \(X_{w,P}\) which are parametrized by W/W(\(\chi)\). For each component in the decomposition he defines a representation \(r_{w,P}\) using Deligne-Lusztig induction, and proves that in the Grothendieck ring of class functions on G, \(Br([\theta (\chi)])=\sum \epsilon_{w,P}r_{w,P}\) where \(\epsilon_{w,P}\) is a well-determined sign and [\(\theta\) (\(\chi)\)] is the class function supported on semisimple elements, which is defined by \(\sum_{w\in W}\chi^ w\) (more precisely the lift via \(\theta\) : \(K^*\to {\mathbb{C}}^*\) of the latter to \({\mathbb{C}})\). This formula is formally equivalent to one for Br(\(\rho\) (\(\chi)\)) where \(\rho\) (\(\chi)\) is the highest-weight module corresponding to \(\chi\).
scientific article; zbMATH DE number 3981432
scientific article